    Theory of differential offset continuation  Next: Comparison with Bolondi's OC Up: Introducing the offset continuation Previous: Proof of validity

## Proof of kinematic equivalence

In order to prove the validity of equation (4), it is convenient to transform it to the coordinates of the initial shot gathers: , , and . The transformed equation takes the form (6)

Now the goal is to prove that any reflection traveltime function in a constant velocity medium satisfies equation (6).

Let and be the source and the receiver locations, and be a reflection point for that pair. Note that the incident ray and the reflected ray form a triangle with the basis on the offset ( ). Let be the angle of from the vertical axis, and be the analogous angle of (Figure 1). The law of sines gives us the following explicit relationships between the sides and the angles of the triangle : (7) (8)

Hence, the total length of the reflected ray satisfies (9)

Here is the reflection angle ( ), and is the central ray angle ( ), which coincides with the local dip angle of the reflector at the reflection point. Recalling the well-known relationships between the ray angles and the first-order traveltime derivatives (10) (11)

we can substitute (9), (10), and (11) into (6), which leads to the simple trigonometric equality (12)

It is now easy to show that equality (12) is true for any and , since ocoray
Figure 1.
Reflection rays in a constant velocity medium (a scheme).    Thus we have proved that equation (6), equivalent to (4), is valid in constant velocity media independently of the reflector geometry and the offset. This means that high-frequency asymptotic components of the waves, described by the OC equation, are located on the true reflection traveltime curves.

The theory of characteristics can provide other ways to prove the kinematic validity of equation (4), as described by Fomel (1994) and Goldin (1994).    Theory of differential offset continuation  Next: Comparison with Bolondi's OC Up: Introducing the offset continuation Previous: Proof of validity

2014-03-26